Petrol kart conversion to electric

gain power? No. Where would that additional power come from. Any gear will reduce the power available at the wheels. Gearing is used to have the motor work in its optimum range of power generation and adapt the motor speed to the wheel rpm.
An electric motor has a wide range of operating speeds and creates torque at very low pm. Therefore electric drives are usually not geared or use only a low number of gears.

Would a starter motor work for 30 minutes or so? I think, but I'm by no means an expert here, starter motors are designed for short term operation only and will overheat easily when operated constantly for a longer time.

At ~ 6kW you will need a 3 kWh battery for 30 minutes operation. Plus some spare kWh as you don't want to drain the battery completely and you'll also want to account for ageing. At 12 V that amounts to 250+ Ah. Quite some battery.
Have you given thought to an electric bicycle conversion kit? You get the motor, the controller and a suitable sprocket to drive a chain. These kits come with or without battery, so you can use your own custom battery. The thing you'll have to watch out for is that you get a kit with manual speed control. What are the regulations in GB? In Germany electric bikes are required to support only while the rider is pedaling. You don't want that kind of controller. You'll want one where speed can be adjusted manually. On a bike that would be a grip on the handlebar. You'll have to adapt that to a foot operated pedal, I think.